CN113376569B - Nested array sparse representation direction-of-arrival estimation method based on maximum likelihood - Google Patents
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
- G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
- G01S3/14—Systems for determining direction or deviation from predetermined direction
- G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
- G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
- G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
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Abstract
The invention discloses a maximum likelihood-based nested array sparse representation direction-of-arrival estimation method, which comprises the steps of firstly calculating echo signals of a receiving array according to the direction of arrival from a target to a nested array and an array structure of the nested array, and further calculating a covariance matrix of the echo signals; dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set; the vectorization covariance matrix is subjected to sparse expansion on the angle set, and a nested array arrival direction estimation sparse model is obtained; constructing a block diagonal matrix, removing noise items in the sparse model, and obtaining a denoised nested array arrival direction estimation sparse model; and then, calculating a noise whitening matrix by combining the covariance matrix, combining the denoised nested matrix arrival direction estimation sparse model, calculating a data model after noise whitening, further calculating grid maximum likelihood estimation, and finally obtaining a arrival direction maximum likelihood estimation value. The invention realizes the estimation performance of the direction of arrival of the nested array under the conditions of low signal to noise ratio and few snapshots.
Description
Technical Field
The invention belongs to the technical field of radars, and particularly relates to a nested array sparse representation direction-of-arrival estimation method based on maximum likelihood.
Background
The direction of arrival estimation technology is an indispensable technical means for realizing target positioning, and the azimuth information of the target in the airspace can be obtained through the direction of arrival estimation result. The traditional typical method has a multiple signal classification method and a rotation invariant subspace method, and the two methods break through the Rayleigh limit, can realize super resolution of a target, but have severely reduced performance in a low signal-to-noise ratio or less snapshot environment.
The maximum likelihood estimation assumes that the signal source is a random process with a known distribution, and uses the known sample result information to extrapolate back the most likely parameter values to be estimated that would result in the occurrence of these sample results. Good estimation performance can still be obtained under the condition of low signal-to-noise ratio or coherent signal source, but solving is difficult, and optimization is required in a multidimensional parameter space, which results in great calculation amount of the algorithm, and the performance of the algorithm depends on the selection of initial values.
The nested array is formed by nesting a plurality of uniform linear arrays, the redundancy of space sampling is reduced through non-uniform arrangement, the degree of freedom of the linear array can be improved, and the closed solution of the array element positions and the degree of freedom which can be improved by the nested array can be calculated through the total number of the array elements.
The basic idea of the sparse representation theory is to replace the basis function set with an atomic dictionary composed of a redundancy function set, so that signals can be represented as linear combinations of a few atomic column vectors in the atomic dictionary. When a small number of point targets are distributed in the airspace, the targets have sparsity for the whole airspace angle, so that the sparse representation theory can be applied to the direction of arrival estimation. But it is assumed that the target falls exactly on the angular grid of the division, which will inevitably present a model mismatch problem.
Aiming at the problem of model mismatch in a sparse representation direction-of-arrival estimation method, the existing method is mainly divided into two main types, one type is a multiple reconstruction method, the calculated amount of an algorithm grows exponentially along with the increase of the number of grids, and the other type is a Bayesian estimation method, and the problem of dictionary matrix grid mismatch is reduced, but the mismatch amount still exists.
Disclosure of Invention
The invention aims to provide a maximum likelihood-based nested array sparse representation direction-of-arrival estimation method, so as to realize the direction-of-arrival estimation performance of a nested array under low signal-to-noise ratio and few snapshots.
The invention adopts the technical scheme that the maximum likelihood-based nested array sparse representation direction-of-arrival estimation method is implemented according to the following steps:
step 1, calculating echo signals of a receiving array according to the direction of arrival of a target to the receiving array of the nested array radar system and the array structure of the nested array;
step 2, calculating a covariance matrix of the received data of the nested array according to the echo signals of the receiving array obtained in the step 1;
step 3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set; the nested array obtained in the vectorization step 2 receives the data covariance matrix and is sparsely unfolded on the angle set to obtain a nested array arrival direction estimation sparse model;
step 4, constructing a block diagonal matrix, and removing noise items in the sparse model in the step 3 to obtain a denoised nested array arrival direction estimation sparse model; and (2) calculating a noise whitening matrix by combining the covariance matrix obtained in the step (2), and calculating a sparse model after noise whitening by combining the obtained denoised nested array direction of arrival estimation sparse model;
and 5, calculating a target direction of arrival according to the sparse model after noise whitening established in the step 4.
The present invention is also characterized in that,
the echo signal y (t) of the receive array in step 1 is calculated as follows:
y(t)＝A(θ)s(t)+n(t)，
wherein s (t) = [ s ] 1 (t),s 2 (t),…,s K (t)] T Representing signal vectors, [] T For transpose operations, K represents the target number, n (t) represents the channel noise vector, assuming compliance with complex Gaussian distribution, i.e Complex gaussian distribution, σ, representing mean μ, covariance matrix Σ 2 Represents noise power, I M×M Represents an identity matrix with dimension M x M, M represents the number of array elements, A (theta) is an array manifold matrix, and A (theta) = [ a (theta) 1 ),a(θ 2 ),…,a(θ k ),…,a(θ K )]，a(θ k ) Representing the array steering vector, θ k Represents the incoming wave direction of the kth object, k=1, 2, …, K, < >>(·) m Represents the mth element of the vector, D m Representing the m-th array element relative reference array of nested array radar systemThe position information of the element, m=1, 2, …, M, λ represents the wavelength of the electromagnetic wave, t represents the time of sample rate normalization, t=1, 2, …, L is the total snapshot number.
Step 2 covariance matrixThe calculation of (2) is as follows:
wherein ( H Is a conjugate transpose operation.
The step 3 is specifically as follows:
step 3.1, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set Θ:
Θ＝{θ 1 ,θ 2 ,…,θ n ,…,θ N }，
wherein N represents the number of division of airspace angles, θ n An angle representing the nth division, n=1, 2, …, N;
step 3.2, vectorizing the covariance matrix of the received data of the nested array obtained in the step 2
Wherein vec (·) represents a vectorization operation;
step 3.3, vectorizing the y obtained in step 3.2 v Sparse expansion is carried out on the angle set theta obtained in the step 3.1:
y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v ，
wherein, represents Kronecker product, (. Cndot.) represents conjugate operation, p Θ The angle information corresponding to the nonzero position is the incoming wave direction of the target, which is the sparse vector, +.>e m Represents a unit vector, Δy, in which the m-th element is 0 and the other elements are 1 v The vector after the difference value vectorization of the nested array theoretical covariance matrix and the actual covariance matrix is represented, and the sparse model for estimating the arrival direction of the nested array is y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v 。
The step 4 is specifically as follows:
step 4.1, constructing a block diagonal matrix J:
wherein J is m ＝[e 1 ,…,e m-1 ,e m+1 ,…,e M ]，m＝2,…,M-1，J 1 ＝[e 2 ,…,e M ]，J M ＝[e 1 ,…,e M-1 ]；
Step 4.2, covariance matrix obtained according to step 2And step 4.1, obtaining a block diagonal matrix J, and calculating a noise whitening matrix W:
step 4.3, establishing a denoised nested array direction-of-arrival estimation sparse model:
removing noise items in the sparse model in the step 3 by using the block diagonal matrix J obtained in the step 4.1 to obtain a denoised nested array arrival direction estimation sparse model:
y J ＝Jy v ＝JΨ(Θ)p Θ +JΔy v ；
step 4.4, establishing a noise whitened arrival direction estimation sparse model y according to the noise whitening matrix W obtained in step 4.2 and the sparse model obtained in step 4.3 w ：
y w ＝W -1/2 y J ＝W -1/2 JΨ(Θ)p Θ +ε＝Φ(Θ)p Θ +ε，
Wherein Φ (Θ) =w -1/2 JΨ(Θ)，Obeys complex gaussian white noise distribution.
The step 5 is specifically as follows:
step 5.1, assume sparse vector p Θ Obeying complex gaussian distribution, i.e
Where P (·|·) represents conditional probability Γ=diag (γ) 1 ，γ 2 ，…，γ N ) Diag (·) represents a diagonal operation;
step 5.2, according to Φ (Θ), y in step 4.4 w And Γ in step 5.1, respectively calculating the sparse vector p Θ Mean of (2)And covariance matrix->
Step 5.3, sparse vector p obtained according to step 5.2 Θ Mean of (2)And covariance matrix->Computing grid maximum likelihood estimate +.>
Wherein ζ is a very small positive number, (. Cndot.) in the form of a square q Represents the q-th iteration, (. Cndot.) the following steps n,n Elements representing the nth row and nth column of the matrix;
step 5.4, all grid maximum likelihood estimates obtained according to step 5.3Forming a spatial spectrum, calculating a covariance matrix Sigma from the spectral peak positions -k ：
Σ -k ＝Φ(Θ -k )diag(γ -k )Φ(Θ -k ) H +I M(M-1)×M(M-1) ，
Wherein,grid corresponding to kth signal source for deleting spatial spectrum peak from set Θ> Indicating the deletion of the k signal source from the grid maximum likelihood estimation result obtained in step 5.3>Is a function of the estimated value of (2);
step 5.5, calculating parameters
The covariance matrix Σ obtained according to step 5.4 k And the parameters obtained in step 5.5Calculating the maximum likelihood estimation theta of the target direction of arrival k ：
Wherein,re {. Cndot. } represents the real part operation, +.>Represents the grid corresponding to the kth signal source +.>Angle set of left and right fields, argmax [ · ]]Representing the variable value taking the maximum value of the function.
Compared with a uniform array, the method for estimating the direction of arrival by using the sparse representation of the nested array based on the maximum likelihood has the advantages that the method for estimating the direction of arrival by using the sparse representation of the nested array based on the maximum likelihood can effectively improve the estimation precision and the resolution performance of an algorithm because the data after the covariance matrix of the vectorized nested array is adopted to build a sparse reconstruction model of the direction of arrival; compared with the prior art, the method has the advantages that the maximum likelihood method is adopted, the maximum likelihood model of the direction of arrival estimation is established, the incoming wave direction of the target can be obtained through one-dimensional search, and the problem of calculation amount surge caused by repeated reconstruction and the problem of Taylor series approximation model mismatch are avoided.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a schematic diagram of a two-layer nested array;
FIG. 3 is a graph of simulation results of spatial spectrum of 9 uncorrelated signals in the airspace using the method of the present invention;
FIG. 4 is a graph of simulation results of a plot of probability of successful resolution with signal to noise ratio for two targets with smaller air angle separation using prior methods and using the methods of the present invention;
fig. 5 is a graph of simulation results of the variation of root mean square error with snapshot number for two objects with larger air angle separation using the prior art method and the method of the present invention.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The invention discloses a maximum likelihood-based nested array sparse representation direction-of-arrival estimation method, which is implemented by a flow chart shown in figure 1 and specifically comprises the following steps of:
and step 1, calculating echo signals of the receiving array according to the arrival direction of the target to the receiving array of the nested array radar system and the array structure of the nested array.
The echo signal y (t) of the receive array in step 1 is calculated as follows:
y(t)＝A(θ)s(t)+n(t)，
wherein s (t) = [ s ] 1 (t),s 2 (t),…,s K (t)] T Representing signal vectors, [] T For transpose operations, K represents the target number, n (t) represents the channel noise vector, assuming compliance with complex Gaussian distribution, i.e Represents mean value μ, covariance matrix ΣComplex gaussian distribution, sigma 2 Represents noise power, I M×M Represents an identity matrix with dimension M x M, M represents the number of array elements, A (theta) is an array manifold matrix, and A (theta) = [ a (theta) 1 ),a(θ 2 ),…,a(θ k ),…a(θ K )]，a(θ k ) Representing the array steering vector, θ k Represents the incoming wave direction of the kth object, k=1, 2, …, K, < >>(·) m Represents the mth element of the vector, D m The M-th array element of the nested array radar system is represented by position information of the M-th array element relative to a reference array element, m=1, 2, …, M and lambda represent wavelengths of electromagnetic waves, t represents time for normalizing sampling rates, and t=1, 2, … and L represent total snapshot numbers.
Step 2, calculating covariance matrix of the received data of the nested array according to the echo signals of the receiving array obtained in the step 1
Step 2 covariance matrixThe calculation of (2) is as follows:
wherein ( H Is a conjugate transpose operation.
Step 3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set; and (3) vectorizing the nested array receiving data covariance matrix obtained in the step (2), and sparsely expanding the data covariance matrix on an angle set to obtain a nested array arrival direction estimation sparse model.
The step 3 is specifically as follows:
step 3.1, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set Θ:
Θ＝{θ 1 ,θ 2 ,…,θ n ,…,θ N }，
wherein N represents the number of division of airspace angles, θ n An angle representing the nth division, n=1, 2, …, N;
step 3.2, vectorizing the covariance matrix of the received data of the nested array obtained in the step 2
Wherein vec (·) represents a vectorization operation;
step 3.3, vectorizing the y obtained in step 3.2 v Sparse expansion is carried out on the angle set theta obtained in the step 3.1:
y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v ，
wherein, represents Kronecker product, (. Cndot.) represents conjugate operation, p Θ The angle information corresponding to the nonzero position is the incoming wave direction of the target, which is the sparse vector, +.>e m Represents a unit vector, Δy, in which the m-th element is 0 and the other elements are 1 v The vector after the difference value vectorization of the nested array theoretical covariance matrix and the actual covariance matrix is represented, and the sparse model for estimating the arrival direction of the nested array is y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v 。
Step 4, constructing a block diagonal matrix, and removing noise items in the sparse model in the step 3 to obtain a denoised nested array arrival direction estimation sparse model; and (3) calculating a noise whitening matrix by combining the covariance matrix obtained in the step (2), and calculating a sparse model after noise whitening by combining the obtained denoised nested array direction of arrival estimation sparse model.
The step 4 is specifically as follows:
step 4.1, constructing a block diagonal matrix J:
wherein J is m ＝[e 1 ,…,e m-1 ,e m+1 ,…,e M ]，m＝2,…,M-1，J 1 ＝[e 2 ,…,e M ]，J M ＝[e 1 ,…,e M-1 ]；
Step 4.2, covariance matrix obtained according to step 2And step 4.1, obtaining a block diagonal matrix J, and calculating a noise whitening matrix W:
and 4.3, establishing a denoised nested array direction-of-arrival estimation sparse model.
Removing noise items in the sparse model in the step 3 by using the block diagonal matrix J obtained in the step 4.1 to obtain a denoised nested array arrival direction estimation sparse model:
y J ＝Jy v ＝JΨ(Θ)p Θ +JΔy v ；
step 4.4, establishing a noise whitened arrival direction estimation sparse model y according to the noise whitening matrix W obtained in step 4.2 and the sparse model obtained in step 4.3 w ：
y w ＝W -1/2 y J ＝W -1/2 JΨ(Θ)p Θ +ε＝Φ(Θ)p Θ +ε，
Wherein Φ (Θ) =w -1/2 JΨ(Θ)，Obeys complex gaussian white noise distribution.
And 5, calculating a target direction of arrival according to the sparse model after noise whitening established in the step 4.
The step 5 is specifically as follows:
step 5.1, assume sparse vector p Θ Obeying complex gaussian distribution, i.e
Where P (·|·) represents conditional probability Γ=diag (γ) 1 ，γ 2 ，…，γ N ) Diag (·) represents a diagonal operation;
step 5.2, according to Φ (Θ), y in step 4.4 w And Γ in step 5.1, respectively calculating the sparse vector p Θ Mean of (2)And covariance matrix->
Step 5.3, sparse vector p obtained according to step 5.2 Θ Mean of (2)And covariance matrix->Computing grid maximum likelihood estimate +.>
Wherein ζ is a very small positive number, (. Cndot.) in the form of a square q Represents the q-th iteration, (. Cndot.) the following steps n,n Elements representing the nth row and nth column of the matrix;
step 5.4, all grid maximum likelihood estimates obtained according to step 5.3Forming a spatial spectrum, calculating a covariance matrix Sigma from the spectral peak positions -k ：
Σ -k ＝Φ(Θ -k )diag(γ -k )Φ(Θ -k ) H +I M(M-1)×M(M-1) ，
Wherein,grid corresponding to kth signal source for deleting spatial spectrum peak from set Θ> Indicating the deletion of the k signal source from the grid maximum likelihood estimation result obtained in step 5.3>Is a function of the estimated value of (2);
step 5.5, calculating parameters
Step 5.6, covariance matrix Σ obtained according to step 5.4 -k And the parameters obtained in step 5.5
Calculating the maximum likelihood estimation theta of the target direction of arrival k ：
Wherein,re {. Cndot. } represents the real part operation, +.>Represents the grid corresponding to the kth signal source +.>Angle set of left and right fields, argmax [ · ]]Representing the variable value taking the maximum value of the function.
The nested array sparse representation wave direction estimation method based on the maximum likelihood can be used for a nested array radar system, and under the condition that the target meets the airspace sparsity, a sparse grid iteration process and an angle fine estimation expression are established through the maximum likelihood method, so that the wave direction estimation performance of the nested array radar system is improved.
The estimation performance of the invention on the target angle information can be further verified through the following simulation.
1. Experimental scenario:
with the two-layer nested array as shown in fig. 2, the total number of array elements is m=6, the array element position sets are {0, d,2d,3d,7d,11d }, d is equal to half wavelength, the space domain angle division interval is 1 °, and the divided angle set is { -90 °:1 °:90 ° }.The initial value is estimated using least squares, i.e. +.>p0＝(Φ(Θ)) + y，(·) + Representing the generalized inverse. The iteration termination condition is that the maximum iteration number is reached or the two iteration updates meet gamma q+1 -γ q || 2 /||γ q || 2 Iota is not more than 2000 times, and gamma= [ gamma ] is the maximum number of iterations 1 ，γ 2 ，…，γ N ] T ，ι＝10- 4 。
2. Experimental content and analysis
Experiment one: the method of the invention is used for carrying out the direction of arrival estimation on 9 uncorrelated equipower signal sources in the far field of space to obtain the spatial spectrum of angle estimation, as shown in figure 3, wherein 'o' represents the direction of arrival of a real target.
As can be seen from fig. 3, the method of the present invention can successfully distinguish the equipower signals from 9 different directions, and is suitable for the scene that the target number is greater than the array element number.
Experiment II: and changing the signal to noise ratio, and carrying out 200 Monte Carlo simulation experiments on each signal to noise ratio by using the prior method and the method of the invention, and respectively counting the change curves of the successful resolution probabilities of the prior method and the method of the invention on two targets with smaller air angle intervals along with the signal to noise ratio, wherein the change curves are shown in figure 4.
As can be seen from fig. 4, as the signal-to-noise ratio increases, the probability of successful resolution of the existing method and the method of the present invention for two targets with smaller air angle interval gradually increases to 100%, but the probability of successful resolution of the method of the present invention at each signal-to-noise ratio is greater than or equal to that of the existing method, which indicates that the resolution performance of the method of the present invention is superior to that of the existing method.
Experiment III: changing the snapshot numbers, carrying out 200 Monte Carlo simulation experiments on each snapshot number by using the prior method and the method of the invention, and respectively counting the change curves of root mean square errors of two targets with larger air angle interval along with the snapshot numbers by using the prior method and the method of the invention, as shown in figure 5.
As can be seen from fig. 5, as the number of shots increases, the root mean square error of the existing method and the method of the present invention for two targets with larger air angle interval gradually decreases, but the root mean square error of the method of the present invention at each number of shots is smaller than or equal to the existing method, especially when the number of shots is smaller than or equal to 40, the root mean square error of the method of the present invention is significantly lower than the existing method, which indicates that the estimation performance of the method of the present invention is better than the existing method.
In conclusion, the method can effectively estimate the direction of arrival of the space domain target, and improves the resolution and estimation performance of the target.
Claims (6)
1. The nested array sparse representation direction-of-arrival estimation method based on the maximum likelihood is characterized by comprising the following steps of:
step 1, calculating echo signals of a receiving array according to the direction of arrival of a target to the receiving array of the nested array radar system and the array structure of the nested array;
step 2, calculating a covariance matrix of the received data of the nested array according to the echo signals of the receiving array obtained in the step 1;
step 3, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set; the nested array obtained in the vectorization step 2 receives the data covariance matrix and is sparsely unfolded on the angle set to obtain a nested array arrival direction estimation sparse model;
step 4, constructing a block diagonal matrix, and removing noise items in the sparse model in the step 3 to obtain a denoised nested array arrival direction estimation sparse model; and (2) calculating a noise whitening matrix by combining the covariance matrix obtained in the step (2), and calculating a sparse model after noise whitening by combining the obtained denoised nested array direction of arrival estimation sparse model;
and 5, calculating a target direction of arrival according to the sparse model after noise whitening established in the step 4.
2. The maximum likelihood-based nested array sparse representation direction of arrival estimation method according to claim 1, wherein the echo signals y (t) of the receiving array in step 1 are calculated as follows:
y(t)＝A(θ)s(t)+n(t)，
wherein s (t) = [ s ] 1 (t),s 2 (t),…,s K (t)] T Representing signal vectors, [] T For transpose operations, K represents the target number, n (t) represents the channel noise vector, assuming compliance with complex Gaussian distribution, i.e Complex gaussian distribution, σ, representing mean μ, covariance matrix Σ 2 Represents noise power, I M×M Represents an identity matrix with dimension M x M, M represents the number of array elements, A (theta) is an array manifold matrix, and A (theta) = [ a (theta) 1 ),a(θ 2 ),…,a(θ k ),…,a(θ K )]，a(θ k ) Representing the array steering vector, θ k Represents the incoming wave direction of the kth object, k=1, 2, …, K, < >>(·) m Represents the mth element of the vector, D m The M-th array element of the nested array radar system is represented by position information of the M-th array element relative to a reference array element, m=1, 2, …, M and lambda represent wavelengths of electromagnetic waves, t represents time for normalizing sampling rates, and t=1, 2, … and L represent total snapshot numbers.
3. The maximum likelihood-based nested matrix sparse representation direction of arrival estimation method of claim 2, wherein the step 2 covariance matrixThe calculation of (2) is as follows:
wherein ( H Is a conjugate transpose operation.
4. The maximum likelihood-based nested array sparse representation direction of arrival estimation method of claim 3, wherein said step 3 is specifically as follows:
step 3.1, dividing the whole airspace in an angle dimension according to a sparse representation theory to obtain an angle set Θ:
Θ＝{θ 1 ,θ 2 ,…,θ n ,…,θ N }，
wherein N represents the number of division of airspace angles, θ n An angle representing the nth division, n=1, 2, …, N;
step 3.2, vectorizing the covariance matrix of the received data of the nested array obtained in the step 2
Wherein vec (·) represents a vectorization operation;
step 3.3, vectorizing the y obtained in step 3.2 v Sparse expansion is carried out on the angle set theta obtained in the step 3.1:
y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v ，
wherein, represents Kronecker product, (. Cndot.) represents conjugate operation, p Θ The angle information corresponding to the nonzero position is the incoming wave direction of the target, which is the sparse vector, +.>e m Represents a unit vector, Δy, in which the m-th element is 0 and the other elements are 1 v The vector after the difference value vectorization of the nested array theoretical covariance matrix and the actual covariance matrix is represented, and the sparse model for estimating the arrival direction of the nested array is y v ＝Ψ(Θ)p Θ +σ 2 1+Δy v 。
5. The maximum likelihood-based nested array sparse representation direction of arrival estimation method of claim 4, wherein said step 4 is specifically as follows:
step 4.1, constructing a block diagonal matrix J:
wherein J is m ＝[e 1 ,…,e m-1 ,e m+1 ,…,e M ]，m＝2,…,M-1，J 1 ＝[e 2 ,…,e M ]，J M ＝[e 1 ,…,e M-1 ]；
Step 4.2, covariance matrix obtained according to step 2And step 4.1, obtaining a block diagonal matrix J, and calculating a noise whitening matrix W:
step 4.3, establishing a denoised nested array direction of arrival estimation sparse model;
removing noise items in the sparse model in the step 3 by using the block diagonal matrix J obtained in the step 4.1 to obtain a denoised nested array arrival direction estimation sparse model:
y J ＝Jy v ＝JΨ(Θ)p Θ +JΔy v ；
step 4.4, establishing a noise whitened arrival direction estimation sparse model y according to the noise whitening matrix W obtained in step 4.2 and the sparse model obtained in step 4.3 w ：
y w ＝W -1/2 y J ＝W -1/2 JΨ(Θ)p Θ +ε＝Φ(Θ)p Θ +ε，
Wherein Φ (Θ) =w -1/2 JΨ(Θ)，Obeys complex gaussian white noise distribution.
6. The maximum likelihood-based nested array sparse representation direction of arrival estimation method of claim 5, wherein said step 5 is specifically as follows:
step 5.1, assume sparse vector p Θ Obeying complex gaussian distribution, i.e
Where P (·|·) represents conditional probability Γ=diag (γ) 1 ，γ 2 ，…，γ N ) Diag (·) represents a diagonal operation;
step 5.2, according to Φ (Θ), y in step 4.4 w And Γ in step 5.1, respectively calculating the sparse vector p Θ Mean of (2)And covariance matrix->
Step 5.3, sparse vector p obtained according to step 5.2 Θ Mean of (2)And covariance matrix->Computing grid maximum likelihood estimate +.>
Wherein ζ is a very small positive number, (. Cndot.) in the form of a square q Represents the q-th iteration, (. Cndot.) the following steps n,n Elements representing the nth row and nth column of the matrix;
step 5.4, all grid maximum likelihood estimates obtained according to step 5.3Forming a spatial spectrum, calculating a covariance matrix Sigma from the spectral peak positions -k ：
Σ -k ＝Φ(Θ -k )diag(γ -k )Φ(Θ -k ) H +I M(M-1)×M(M-1) ，
Wherein,grid corresponding to kth signal source for deleting spatial spectrum peak from set Θ> Indicating the deletion of the k signal source from the grid maximum likelihood estimation result obtained in step 5.3>Is a function of the estimated value of (2);
step 5.5, calculating parameters
Step 5.6, covariance matrix Σ obtained according to step 5.4 -k And the parameters obtained in step 5.5Calculating the maximum likelihood estimation theta of the target direction of arrival k ：
Wherein,re {. Cndot. } represents the real part operation, +.>Represents the grid corresponding to the kth signal source +.>Angle set of left and right fields, argmax [ · ]]Representing the variable value taking the maximum value of the function.
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